Method and an apparatus for searching or comparing sites using routes or route lengths between sites and places within a transportation system

ABSTRACT

Embodiments relate to searching or comparing sites. One embodiment is a real estate search-or-compare method based on commute durations. The method efficiently processes public transportation and real estate property data to compute the durations of travel between the real estate properties and the vehicle stops. These durations are stored. A request framework is introduced that allows to express a wide range of search-or-compare requests. During request processing, the method identifies parts of the commute paths that depend on any real estate property. Because durations for these parts were precomputed and stored, the method can determine commute durations to every real estate property in a scalable manner. As a result, the method rapidly responds to requests within the real estate market of one of the largest metropolitan areas in existence today. Other embodiments include: searching or comparing based on a monetary cost, transportation using private cars, and sites other than real estate properties. A computer system and a computer service also embody the method.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based upon, and claims the priority dates of,applications:

[Country] [Application Number] [Filing Date] USA 62/632,419 Feb. 20,2018 USA 62/758,710 Nov. 12, 2018 USA 62/780,268 Dec. 16, 2018 USA62/800,428 Feb. 2, 2019,which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

The present invention relates to searching or comparing sites. Atraditional goal of searching is to find a site, from among a range ofpossible alternatives, that achieves an optimization objective, such asminimize a route length given specific travel requirements and desiredfeatures of the sought site. For example, when searching for real estateproperties given required destinations of commutes and real estateproperty features, a goal may be to enumerate real estate propertieswith matching features that have the shortest commute durations. Othergoal may be to compare any real estate properties using commutedurations.

BRIEF SUMMARY OF THE INVENTION

Embodiments include a method for searching or comparing sites, acomputer system that implements and executes the method, and a computerservice that receives search or compare requests from users and respondswith site and route information.

According to an embodiment of the present invention, a method forsearching or comparing sites using routes or route lengths is provided.The method uses extensive preprocessing to precompute and store in adatabase routes or route lengths between each site and representativeswithin a transportation system. The method introduces asearch-or-compare framework for sites. When a request containing a routespecification is received, precomputed data is retrieved from thedatabase to rapidly compute a route or a route length for each site.Sites may be searched or compared using routes or route lengths.

According to an embodiment of the present invention, a computer systemfor searching or comparing sites using routes or route lengths isprovided. The system is a combination of hardware and software. Itobtains data about a transportation system and sites from one or moredata providers. The system builds graphs that model travel between thesites and representatives within the transportation system. The systemcomputes graph paths, and stores graph paths or graph path lengths. Thisenables to quickly compute routes or route lengths for every site when arequest is received, and search or compare sites using routes or routelengths.

According to an embodiment of the present invention, a computer servicefor searching or comparing sites using routes or route lengths isprovided. The service allows the user to specify a search or comparerequest through a User Interface on a device, for example a smartphone.The request contains a route specification and a filtering condition. Inresponse, the service presents sites that match the filtering conditionalong with routes or route lengths for the matched sites, or the servicecompares sites using routes or route lengths.

It is fundamentally necessary to quickly compute a route length for eachsite. We sketch a mathematical proof. Consider any search-or-comparemethod

. We can design an adversarial request that will force

to respond with any given ordered list of sites. The adversary has twomechanisms at its disposal: (1) provide a request with a routespecification that creates an order of sites with respect to the routelength, the order selected by the adversary, and (2) provide a requestwith a filtering condition on sites that matches a subset of sites thatis selected by the adversary. Thus the

must respond with an ordered list of sites arbitrarily selected by theadversary at request time. Details of the proof sketch are outside ofthe scope of a patent application.

The method, the computer system, and the computer service each jointlyperforms tasks that are not simply generic nor well-understood by priorart. Prior art includes: U.S. Pat. No. 8,417,409 B2; continuation U.S.Pat. No. 8,738,286 B2; division U.S. Pat. No. 8,756,014 B2; KR10-1692501 B1; continuation PCT/KR2016/01083; WO 2017/065431 A1; US2018/0232824 A1; and KR 10-1905593 B1.

The embodiments of the invention presented here are for illustrativepurpose; they are not intended to be exhaustive. Many modifications andvariations will be apparent to those of ordinary skill in the artwithout departing from the scope and spirit of the embodiments.

In the presentation, the terms “the first”, “the second”, “the”, andsimilar, are not used in any limiting sense, but for the purpose ofdistinguishing, unless otherwise is clear from the context. Anexpression in a singular form includes the plural form, unless otherwiseis clear from the context. The terms “having”, “including”,“comprising”, and similar, indicate an existence of components orfeatures, and do not preclude other components or features from existingor being added.

BRIEF DESCRIPTION OF DRAWINGS

The drawings included in the invention disclosure exemplify variousfeatures and advantages of the embodiments of the invention:

FIG. 1: depicts an example color rendering of travel durations ofcommute paths according to an embodiment of the invention, legend: “Anexample color rendering of travel durations of commute paths for realestate properties with two commute paths:property→geo.0.0→geo.0.1→property, and property→geo.1.0→property. Weused a map tile engine by Google, any other may be used instead.”;

FIG. 2: depicts an example process flow of preprocessing and storing ofdata according to an embodiment of the invention, legend: “An exampleprocess flow of preprocessing and storing of data.”;

FIG. 3: depicts an example process flow of responding to a request usingpreprocessed stored data according to an embodiment of the invention,legend: “An example process flow of responding to a request usingpreprocessed stored data.”;

FIG. 4: depicts example commute paths according to an embodiment of theinvention, legend: “Example commute paths.”;

FIG. 5: depicts example extensions of a public transportation systemgraph for precomputing shortest graph paths according to an embodimentof the invention, legend: “Example public transportation system graphextensions for precomputing shortest graph paths.”;

FIG. 6: depicts an example table/vector storage of precomputed shortesttravel durations according to an embodiment of the invention, legend:“An example table/vector storage of precomputed shortest traveldurations.”;

FIG. 7: depicts an example of a commute path decomposition according toan embodiment of the invention, legend: “An example commute pathH→W₁→W₂→W₃→W₄→H decomposed into a home-dependent part (green) and ahome-independent part (black), illustrated on three homes H₁, H₂, andH₃.”;

FIG. 8: depicts an example of computing the shortest travel duration atthe start of a commute path according to an embodiment of the invention,legend: “An example of computing the shortest travel duration at thestart of a commute path.”;

FIG. 9: depicts an example of computing the shortest travel duration atthe end of a commute path according to an embodiment of the invention,legend: “An example of computing the shortest travel duration at the endof a commute path.”;

FIG. 10: depicts an example pseudocode for computing the shortest travelduration at the start of a commute path according to an embodiment ofthe invention, legend: “An example pseudocode for computingPathFromHomeDurations(→W).”;

FIG. 11: depicts an example flowchart of a computer system according toan embodiment of the invention, legend: “An example flowchart of acomputer system for: precomputing and storing data in a database, andprocessing requests using data retrieved from the database.”;

FIG. 12: depicts an example user request and a response by a computerservice on a smartphone of the user according to an embodiment of theinvention, legend: “An example user request and a response by a computerservice on a smartphone of the user.”.

The drawings are for illustrative purpose only. Other drawings canexemplify the invention without departing from the principles of theinvention, as will be readily recognized by one of ordinary skill in theart.

DETAILED DESCRIPTION OF THE INVENTION 4. Detailed Description

The invention concerns a general case of searching or comparingarbitrary sites using an arbitrary optimization objective that usesroutes or route lengths to arbitrary places. However, for the sake ofease of presentation, we first illustrate the invention through specificsites that are real estate properties, specific places that areworkplaces, and a specific optimization objective of minimizing traveldurations of commutes between real estate properties and workplaces.This illustration is not limiting. In later sections, we explain how themethod works in the general case.

4.1 Real Estate Properties and Commute Paths

Finding a home is a complex endeavor. People spend a significant effortof time and money on a search. Technology has come to help, however.There are several services available online that aggregate real estateproperty listings, and allow people to search for real estate propertieswith specific features, for example a price, a geographical location, anumber of bedrooms, and so on, in the convenience of a web browser or asmartphone. This search yields a shortlist that the person thentypically inspects in flesh.

Location is arguably the most important feature of any real estateproperty, as attested by the “location, location, location” slogan ofreal estate agents. Our invention concerns this feature.

Residents need to commute to work, school, or other places. The traveldurations of these commutes bear on the value of the real estateproperty personalized for its specific residents. In addition, there ismuch global economic value in reducing the travel durations for allresidents of a metropolitan area, in terms of time that can be allocatedto more productive activities, energy that does not need to be spent onmoving people around, etc. Our invention helps deliver these personaland global values.

As a simple illustration, consider a family of two who live in thecapital of South Korea. One person is a government employee at the CityHall, and the other person is a librarian at the Main Library of SeoulNational University. They currently live in a two-bedroom one-bathroomapartment of size 69 square meters that requires a deposit payment of350,000,000 KRW. The apartment is near latitude and longitude (37.5333,127.0746). Their combined daily roundtrip commute lasts 1 h 41 m (about25 minutes each way, on average). However, it turns out that they candecrease the travel duration to just 1 h 16 m (25 minutes savings) aftermoving to an apartment with the same features, but located near(37.5041, 126.8888). The shortest travel duration for any apartment is50 minutes. However, that apartment has other price and size.(Illustration as of Feb. 19, 2018.)

Why is it hard to deliver these improvements for all? A naive approachis to consider every real estate property available on the market thathas the size and other features required by the family, and, given thework places, compute the travel durations by querying any existingonline routing engine. This approach does not scale, however. Oneproblem is the high number of real estate properties available on themarket in a modern metropolitan area. Other problem is the high numberof families/users who may want to seek improvements. Even if we assumethat we can query a routing engine cheaply, the quadratic nature of theproblem still makes the queries expensive in aggregate.

How can we deliver these improvements for all? Our invention explains.It includes the following components:

-   1. The invention defines a model for a commute path. The model is    versatile to cover a wide range of commute paths that occur in    practice, for example going to school, then to a piano class, and    then returning home. The utility of our model is enhanced by our    ability to quickly find real estate properties that minimize travel    durations.-   2. The invention teaches an optimization method that rapidly    computes travel durations. The invention identifies the parts of any    commute path that are dependent on any real estate property. Travel    durations for these parts are precomputed and stored. As a result,    when travel durations need to be found for a commute path, the    invention can rapidly assemble time parts to produce a travel    duration for every real estate property.-   3. An embodiment of the invention is a live computer service. The    service enables the 25 million residents of the Seoul Metropolitan    Area to search or compare real estate properties using travel    durations.

4.2 Method Outline

We use the term travel in a broad sense that includes moving objects ordata. A description of travel is anything that a person of ordinaryskill in the art would name so. Here are some examples of a descriptionof travel: (1) “hey buddy, you need to go one block north, and then turnslightly left”, and (2) “5 dollars”. We may use a term travel path whenwe mean a description of travel. A length of travel is a numeric valuethat a person of ordinary skill in the art can associate with travel,for example a distance, a monetary cost, etc. As other example, we mayuse a term travel duration when we mean a length of travel thatrepresents time. A length of travel is by itself a description oftravel. A description of travel: may not include any length of travel,may include only a length of travel, or may also include some otherdata.

We illustrate some capabilities of the method and introduce terminologysubsequently used. The method can compute travel durations of commutepaths for every real estate property. FIG. 1 illustrates this. Themetropolitan area is colored with squares. The colors represent how longit takes to commute from a real estate property in each specific areausing public transportation: green means short commute, yellow longer,and red longest. A commute path will start at a property, then visitspecific places geo, . . . , and finally return to the property. In asense the property is a free variable, while the geo, . . . are fixed.In FIG. 1, there are two commute paths: an “open-jaw” commute pathproperty→geo.0.0→geo.0.1→property twice per week, and a “roundtrip”commute path property→geo.1.0→property three times per week. We can seethat given these two commute paths and their frequencies, the realestate properties with small weighted sum of shortest travel durationsform an irregular patch (dark green), which may not come as a surprise,given possibly intricate public transportation routes.

At a high level, the method is composed of two parts. The first partcomputes travel durations between real estate properties andrepresentatives that are vehicle stops of a transportation system. Thesetravel durations are stored in a database, so that they can be readilyretrieved when a request is received. See FIG. 2 for an illustration.The second part processes requests. A request contains a commute pathand desired features of a real estate property. When a request isreceived, appropriate travel durations are retrieved from the database,and used together with other data to produce a travel duration for thecommute path for every real estate property that has the desiredfeatures. See FIG. 3 for an illustration. Details and variants of thisoutline are described in subsequent sections.

4.3 Commute Paths

Our method handles a broad range of commutes that people perform withina metropolitan area. A commute starts at a site H that we call a home.The H is an arbitrary location. It can be any real estate property, forexample an apartment, a rented room, a house with a garden, a ranch, ahotel, etc. It can also be a site where a person works, a restaurant, ashop, etc. However, as a naming convention, we use the phrase “home” inmost of the disclosure; this convention is not limiting. In oneembodiment, a commuter travels to various places, and then returns tothe site H.

In one embodiment, a commute lasts within one day, for example acommuter departs from H in the morning and returns back to H in theevening of the same day. In other embodiment, a commute lasts withinmore than one day, for example if a person works a night shift, or has awork shift of 25 hours. In one embodiment, any travel may start at aspecific time, or may end at a specific time, for example 8:12 AM. Inother embodiment, any travel may begin, or end, within a range of time,for example “in the morning”.

In a simplest form, a commuter travels from H to a place W, that we callwork. The W is an arbitrary location. It includes a school, agrandparents home, a weekend golf course, a favorite restaurant, adoctor's office, a place of worship, etc. It can also be a place where aperson lives. However, as a naming convention, we use the phrase “work”in most of the disclosure; this convention is not limiting. Then thecommuter returns from W to H. We call these two travels a roundtripcommute path. See FIG. 4A for an illustration.

An open-jaw commute path is an example of a more complex commute. SeeFIG. 4B for an illustration. Here the commuter travels from H to a placeW₁. Then the commuter travels from the place W₁ to other place W₂.Finally, the commuter travels from the place W₂ back to the site H. Oneexample is when a person travels to a school and then to a piano class.

In general, our method allows for arbitrary travels. In FIG. 4C, we seean illustration of a commute path with a missing travel from W₄ to W₂,and a repeated travel from W₂ to W₃. In FIG. 4D, we see illustrations of“open” commute paths: a commute path that departs from H, but does notreturn to H; and a commute path that returns to H, without previouslydeparting form H. In FIG. 4E, we see an illustration of a “disconnected”commute path. A commute path could start at a home, but end at otherhome. Our method defines a commute path as follows.

Definition 1 A commute path is a collection of travels W₂→W₃→W₄→W₅, . .. , W_(k−2)→W_(k−1), for k≥2 that is even, together with H_(first)→W₁ orW_(k)→H_(last) (so a commute path always contains at least one home andat least one work), that occur at arbitrarily moments of time.

A commute path can be viewed as a specification of a route within atransportation system. The various W and H in Definition 1 specify wherethe commuter wishes to travel.

In one embodiment, we consider a simpler commute path H→W₁→W₂→ . . .→W_(k)→H, for any k≥1, that has shared endpoints between travels, andthe same home at the start and the end. We use this simpler form in mostof the disclosure, because it is common in practice and simplifies ourpresentation. However, it will be obvious to anyone of ordinary skill inthe art that our method applies to our (general) definition of a commutepath.

Our method finds a shortest travel duration along any commute path. Thisby itself has been addressed by various prior art. However, our methodfinds travel durations for all homes in a dramatically reduced amount oftime.

4.4 Preprocessing of Transportation System

The method preprocesses data about public transportation system, so asto precompute and store shortest travel for all homes.

4.4.1 Computation of Shortest Travel

We describe a method for efficiently computing shortest travel betweenall homes, and all locations of public transportation stops. We call thestops stopstations in most of the disclosure, and they include busstops, subway stations, or both.

The method starts with an arbitrary public transportation system graphGT that models a public transportation system, and that can be obtainedfrom prior art. The graph may contain vertexes that represent bus stops,subway stations, or both. Other vertexes may exist in the graph, forexample representing a stop or a turn of a vehicle, or a stop or a turnof a walk. Vertexes in the graph that correspond to stops of vehiclesare denoted bySTOPSTATION_s,indexed by s. There are directed weighted edges, each representing atravel duration along a segment from the source vertex of the edge tothe target vertex of the edge. Other edges may exists in the graph. Anedge may represent a transfer from one vehicle to other vehicle. Thegraph may contain data about departure times or arrival times of variouspublic transportation vehicles. A Dijkstra's shortest graph pathsalgorithm, or an A* (A star) search algorithm, is commonly used tocompute a graph path with a shortest graph path length (sum of weights),representing a shortest travel duration, from aSTOPSTATION_s′to otherSTOPSTATION_s″,for any s′ and s″, also if travel starts at a specific time, or ends ata specific time.

Our method extends the public transportation system graph GT. See FIG. 5for an illustration.

The first extension introduces clusters of stopstations. The methodclusters stopstations using any clustering algorithm; in one embodimentthe method puts two stopstations into one cluster when the geographicaldistance between the two stopstations is at most a threshold, forexample 5 meters, or when a travel duration between the two stopstationsis at most a threshold. We may refer to the location of a cluster, whenwe mean a geographical location within the cluster, for example thecenter of the cluster. For each stopstation cluster c, the method addstwo vertexesSTOPSTATION_CLUSTER_SOURCE_candSTOPSTATION_CLUSTER_TARGET_c,and edges that connect the cluster with stopstations of GTSTOPSTATION_CLUSTER_SOURCE_c→STOPSTATION_s′labeled FirstWaitGetOn, andSTOPSTATION_s′→STOPSTATION_CLUSTER_TARGET_clabelled Zero, for any s′, whenever s′ is in cluster c. The edges havezero weight. The resulting graph is denoted by GC, having vertexes VCand edges EC. A subset of the vertexesSTOPSTATION_CLUSTER_SOURCE_c,for all c, is denoted by VS.

The seconds extension introduces clusters of homes. The method clustershomes using any clustering algorithm; in one embodiment the method usesthe same algorithm as when clustering stopstations. Similarly, we mayrefer to the location of a home cluster. For each home cluster s, themethod adds a vertexHOME_CLUSTER_SOURCE_sand a vertexHOME_CLUSTER_TARGET_s.The method connects each home cluster to stopstation clusters usingwalks. Specifically, the method adds an edgeHOME_CLUSTER_SOURCE_s→STOPSTATION_CLUSTER_SOURCE_clabelled Walk, if there is a walk from the location of the home clusters to the location of the stopstation cluster c, for any s and c, withedge weight set to the duration of the walk; and a “reversed” edgeSTOPSTATION_CLUSTER_TARGET_c→HOME_CLUSTER_TARGET_tlabelled Walk when there is a walk in the reverse direction from thelocation of the stopstation cluster c to the location of the homecluster t, for any c and t, with edge weight set to the duration of thewalk. In one embodiment, the method limits walks to shortest walks at aspecific speed, for example 4 km/h, lasting at most a threshold, forexample 1 hour. The resulting graph is denoted by G. We denote by VH theset of vertexesHOME_CLUSTER_SOURCE_s,for all s; and denote by EH the set of edgesHOME_CLUSTER_SOURCE_s→STOPSTATION_CLUSTER_SOURCE_c,for all s and c.

We are interested in computing a shortest graph path from eachstopstation cluster to each home cluster, and a shortest graph path inthe reverse direction from each home cluster to each stopstationcluster.

We remark that clustering allows our method to significantly improve theperformance of a shortest graph paths computation, by “unifying”locations that are essentially the same with respect to shortest graphpaths. For example, in a tall apartment complex, there may be hundredsof homes, and our method will “unify” them into just one home cluster.Thus a shortest graph paths algorithm needs to compute shortest graphpaths for just one home cluster, rather than for hundreds of constituenthomes.

A standard application of a Dijkstra's algorithm still yields poorperformance, however. That application runs the algorithm from eachSTOPSTATION_CLUSTER_SOURCE_cin the G without anyHOME_CLUSTER_SOURCE_svertexes, and then runs the algorithm from eachHOME_CLUSTER_SOURCE_sin the G without anyHOME_CLUSTER_TARGET_tvertexes. The combined asymptotic time complexity isO(|VS|·(|EC|+|EH|+(|VC|+|VH|)log(|VC|+|VH|)))+O(|VH|·(|EC|+|EH|+(|VC|+|VH|)log(|VC|+|VH|))).

Our method improves on this standard application. We observe that in alarge metropolitan area it is often the case that the number of homeclusters is significantly larger than the number of stopstation clusters|VH|»|VS|, given the same clustering threshold. Utilizing thisobservation, our method uses a different algorithm for computingshortest graph paths from home clusters: the method reverses the edgesof G, and for eachSTOPSTATION_CLUSTER_TARGET_cin the reversed graph, computes shortest graph paths to allHOME_CLUSTER_SOURCE_svertexes using Dijkstra's algorithm (anyHOME_CLUSTER_TARGET_tvertexes can be removed). This yields the desired effect, because whenwe reverse the edges of any shortest graph path fromSTOPSTATION_CLUSTER_TARGET_ctoHOME_CLUSTER_SOURCE_sin the reversed graph, we obtain a shortest graph path fromHOME_CLUSTER_SOURCE_stoSTOPSTATION_CLUSTER_TARGET_cin the original (not reversed) graph. Hence, the asymptotic timecomplexity of our method is justO(|VS|·((|EC|+|EH|)+(|VC|+|VH|)log(|VC|+|VH|))).In practice, our method offers a significant reduction in the totalrunning time on a graph G of the Seoul Metropolitan Area.

The method uses a symmetric algorithm in the opposite case when thereare more stopstation clusters than home clusters |VH|<|VS|. In thatcase, the method reverses the edges when computing shortest graph pathsfrom stopstation clusters to home clusters.

In one embodiment, the method uses departure times. The method computesshortest graph paths from eachSTOPSTATION_CLUSTER_SOURCE_cto everyHOME_CLUSTER_TARGET_t,for a given departure time from theSTOPSTATION_CLUSTER_SOURCE_c.In that case, the method uses an appropriate prior art graph GT thatallows to specify the departure time from stops of public transportationvehicles; this case is sometimes represented using a vertex thatcorresponds to a time and a geographical location, and an edge thatrepresents a travel duration starting from the geographical location atthe time. Similarly, the method computes in the reversed graph based ona given arrival time atSTOPSTATION_CLUSTER_TARGET_c.Shortest graph path lengths convert these arrival times to the departuretimes from eachHOME_CLUSTER_SOURCE_s.A similar graph is used to compute shortest graph paths given an arrivaldeadline. An appropriately constructed graph can be used to compute aprobability of arrival before a deadline.

In one embodiment, we do not extend GT with any vertexesSTOPSTATION_CLUSTER_SOURCE_cnor any vertexesSTOPSTATION_CLUSTER_TARGET_c.Instead, we connect any vertexesHOME_CLUSTER_SOURCE_sand any vertexesHOME_CLUSTER_TARGET_tby direct Walk edges with any vertexesSTOPSTATION_s′.In one embodiment, we do not extend GT with any vertexesHOME_CLUSTER_SOURCE_snor any vertexesHOME_CLUSTER_TARGET_t.Instead, we add vertexesHOME_s,each representing a home, that we connect to any vertexesSTOPSTATION_s′directly by Walk edges.

In one embodiment, we do not cluster homes.

In one embodiment, we do not cluster stopstations.

In one embodiment, we compute shortest graph paths given an arrivaldeadline: from stopstation clusters to home clusters given an arrivaldeadline at each home cluster, or from home clusters to stopstationclusters given an arrival deadline at each stopstation cluster.

In one embodiment, we use any shortest graph paths algorithm other thana Dijkstra's algorithm, for example an A* (A star) search algorithm. Inone embodiment, we use an approximation algorithm for shortest graphpaths. We may use an algorithm without any of the performanceimprovements.

In one embodiment, the weights on some graph edges represent monetarycosts of travel, instead of travel durations. Then our method searchesor compares homes based on the monetary cost of commute paths. Any othersemantic of edge weights can be used, for example: a number of transfersamong public transportation vehicles, a duration of wait, a monetarycost of wait, or a distance of travel.

In one embodiment, we apply a multi-objective optimization search basedon a multi-dimensional cost. For example, we search for a shortest graphpath whose length represents a travel duration, such that a monetarycost that represents the graph path is at most a threshold, or when amonetary cost is added as a penalty to the graph path length.

In one embodiment, a travel path is represented by a graph path. Wecompute various features of a shortest travel path using a shortestgraph path, for example: the first (i.e., boarding) or the last (i.e.,disembarking) stopstation of the shortest graph path, the main transitstopstation of the shortest graph path, the number of vehicle transfers,the vehicle where most time is spent during travel (for example bus1234), the total wait time at stopstations, the total walk distance,whether the specific shortest graph path is typically congested duringrush hours, or a sequence of geographical locations along the graphpath. These features may be used during request processing to filtertravel paths whose features match a condition specified in the request.

In one embodiment, we compute two or more graph paths between some graphvertexes. For example, one graph path from a vertex u to a vertex vcould be with at most one bus, and other graph path from the vertex u tothe vertex v could be without any subway, but no more than 10 minuteslonger than a shortest graph path.

In one embodiment, we compute graph paths that match various filteringconditions. For example, graph paths that have at most one transfer, orat most a specific walk duration.

In one embodiment, we use a routing engine (for example prior art) tocompute shortest travel between homes and public transportation stops.Thus we may sometimes not use the graphs described in the currentSection 4.4.1.

The method described in Section 4.4.1 computes a shortest travel fromeach home to each stopstation, and a shortest travel in the reversedirection. Next we describe a data structure for efficient storage andprocessing of shortest travel data used by our method.

4.4.2 Storage of Shortest Travel

In one embodiment, our method stores travel durations in a vector form.The method sequences stopstation clusters as s₁, . . . , s_(n), andsequences home clusters as h₁, . . . , h_(m) in some orders, for examplerandom orders. In one embodiment, these sequences become fixed. For eachs_(i), the method stores a vector of shortest travel durations froms_(i) to the home clusters using the home cluster sequencev _(i)=(t _(i,1) ,t _(i,2) , . . . ,t _(i,m)),so that the value of the vector v_(i) at a coordinate j, v_(i)[j], ist_(i,j) equal to the shortest travel duration from the stopstationcluster s_(i) to the home cluster h_(j). See FIG. 6 for an illustration.

The home cluster sequence simplifies the calculation of a shortesttravel duration to every home cluster. For example, if a commuter needsto travel from a stopstation cluster s_(i) ₁ , and in addition from astopstation cluster s_(i) ₂ , we simply add vectors v_(i) ₁ and v_(i) ₂coordinate-wise, and the resulting sum contains the total travelduration from both stopstation clusters to each home cluster h₁, . . . ,h_(m).

In one embodiment, the method stores durations of reversed travel, fromhomes to stopstations, using the same home cluster sequence. That is,the method stores a vectorv′ _(i)=(t′ _(i,1) ,t′ _(i,2) , . . . ,t′ _(i,m)),so that v′_(i)[j]=t′_(i,j) is the shortest travel duration from the homecluster h_(j) to the stopstation cluster s_(i) (note the transposition;the vector v′_(i) is for one stopstation cluster, even though travel isin the opposite direction).

In general, the two vectors for the same stopstation cluster s_(i) arenot equal, v_(i)≠v′_(i), because the minimum travel duration to a homemay be different than from a home (travel is not symmetric in general).However, in one embodiment the method computes and stores just one ofthe two vectors, and uses it instead of the other, which can save timeand space. In other embodiment, the method stores a coordinate-wiseweighted average of the two vectors, which may decrease a worst-caseerror. The weights may be set to 0.5, or may favor travel from, or mayfavor travel to a specific home cluster, for example based on requestfrequency.

In one embodiment, the method stores travel durations rounded to theclosest minute, using one byte of computer memory, represented as anunsigned integer from 0 to 254, with 255 denoting an unknown or toolarge travel duration. This storage enables efficient vector additionusing modern computer hardware support for Vector Operations andSaturation Arithmetic (for example the AVX-512 instruction set, or GPUintrinsics), while keeping the error at a practically acceptable levelof at most half a minute, and covering common travel durations up toover 4 hours. Any other rounding can be used, for example a duration inseconds can be divided by 120, and rounded to an integer, which willrepresent durations at a 2-minute granularity.

In one embodiment, the method stores a vector not for clusters. Forexample, some of the s₁, . . . , s_(n) represent stopstations (notstopstation clusters), or some of the h₁, . . . , h_(m) represent homes(not home clusters).

In one embodiment, the method uses other form of a vector to storetravel durations. For example, a hash map, or a (coordinate, value)list. These sparse forms may be advantageous when there are many unknownor too large travel durations.

In one embodiment, the method stores travel paths using vectors thatfollow the home cluster sequence h₁, . . . , h_(m), or using other formof a vector, for example a hash map or a (coordinate, value) list. Thesetravel paths may be used during request processing to filter travelpaths whose features match a condition specified in the request.

4.5 Travel Duration of One Commute Path

We describe how our method efficiently computes the shortest travelduration of any commute path. Consider any commute path from a homecluster H, through k≥1 works, back to the home cluster: H→W₁→W₂→ . . .→W_(k)→H. We want to find the shortest travel duration of this commutepath for every H. This may be expensive, because the number of homeclusters may be rather large, for example 500,000 given a clusteringradius of a few meters. However, our method introduces a technique thatdramatically accelerates the search: the method decomposes any commutepath into a home-independent part, and a home-dependent part, asillustrated in FIG. 7.

4.5.1 Middle Part: Travel W₁→W₂→ . . . →W_(k)

The method finds the shortest travel duration for travel W₁→W₂→ . . .→W_(k) that excludes the home cluster. For each segment W_(i)→W_(i+1),the method queries a routing engine (for example prior art) thatcomputes the shortest travel duration from W_(i) to W_(i+1) using thetransportation system (for example, involving walks, subway and busrides, transfers, including a direct walk from W_(i) to W_(i+1)). Thenthe method adds up the shortest travel durations across the different i.The resulting sum is denoted by

${{PathNonHomeDuration}\mspace{14mu}\left( W_{1}\rightarrow\left. W_{2}\rightarrow\left. \ldots\rightarrow W_{k} \right. \right. \right)} = {\sum\limits_{1 \leq i \leq {k - 1}}\;{{RoutingEngineShortestDuration}\mspace{14mu}{\left( W_{i}\rightarrow W_{i + 1} \right).}}}$We note that only k−1 queries to the routing engine are needed. Thisnumber is independent of the number of homes.

In one embodiment, the method uses a departure time, an arrivaldeadline, or other parts of a request, when querying the routing engine.

Next, the method computes the shortest travel duration for the open-jawpart of the commute path: the two segments H→W₁ and W_(k)→H that involvethe home cluster. This computation needs to be especially quick, becauseits result is needed for every home cluster.

4.5.2 Start Part: Travel H→W₁

The method considers two ways of getting from H to W₁. See FIG. 8 for anillustration. The first way is a direct walk. When H and W₁ are nearby,a true shortest travel will often be a direct walk. The method thusqueries a walk engine (for example prior art) to compute the shortestwalk duration walk(H→W₁). The second way uses the transportation system.The method finds the stopstation clusters that are within a thresholddistance, for example 2000 meters, from W₁, denoted by a set A. The setA is a subset of {s₁, . . . , s_(n)} of all stopstation clusters (inFIG. 8 the A={s₁, s₂, s₃}). The method queries a walk engine to retrievea shortest walk duration walk(s_(i)→W₁), for each s_(i) in A. When H andW₁ are not nearby, a true shortest travel will often pass through astopstation in the set A, and then continue along a corresponding walk.Due to the two-way approach we apply, the resulting travel durationoften is a shortest travel duration.

The two-way approach can be applied to each home cluster. We recall thata vector v′_(i) contains shortest travel durations to stopstationcluster s_(i) from consecutive home clusters. Therefore, the methodcomputes a shortest travel duration from a home cluster h_(j) byretrieving vectors v′_(i) from the database, and using the followingformula

$\begin{matrix}{a_{j} = {\min{\left\{ {{{walk}\left( H_{j}\rightarrow W_{1} \right)},{\min\limits_{s_{i} \in A}\left\{ {{\upsilon_{i}^{\prime}\lbrack j\rbrack} + {{walk}\left( s_{i}\rightarrow W_{1} \right)}} \right\}}} \right\}.}}} & \left( {{Equation}\mspace{14mu} 1} \right)\end{matrix}$

In one embodiment, the method computes walk(H_(j)→W₁) only when thedistance from home cluster H_(j) to W₁ is below a threshold, for example2000 meters, or when a travel duration from home cluster H_(j) to W₁ isbelow a threshold. Let us denote the set of such home clusters by J.

In one embodiment, because of the vector representation of the v′, themethod jointly computes the travel duration from every home cluster,using vector operations on the v′ as follows:

for  j = 1  to  m a_(j) = ∞ for  all  s_(i) ∈ A w = walk(s_(i) → W₁)$\begin{pmatrix}a_{1} \\a_{2} \\\vdots \\a_{m}\end{pmatrix} = \begin{pmatrix}{\min\left\{ {a_{1},{{\upsilon_{i}^{\prime}\lbrack 1\rbrack} + w}} \right\}} \\{\min\left\{ {a_{2},{{\upsilon_{i}^{\prime}\lbrack 2\rbrack} + w}} \right\}} \\\vdots \\{\min\left\{ {a_{m},{{\upsilon_{i}^{\prime}\lbrack m\rbrack} + w}} \right\}}\end{pmatrix}$ for  all  j ∈ J w = walk(H_(j) → W₁)a_(j) = min {a_(j), w}.

In one embodiment, the method uses a mathematical formula

${\min\limits_{s_{i} \in A}\left\{ {{{walk}\left( s_{i}\rightarrow W_{1} \right)} + \upsilon_{i}^{\prime}} \right\}},$where the “+” operation adds a number to a value at each coordinate of avector, and the “min” operation computes a minimum value at eachcoordinate across several vectors.

The vector (a₁, . . . , a_(m)) of travel durations is denoted byPathFromHomeDurations(→W ₁)=(a ₁ , . . . ,a _(m)).

An example pseudocode for computing PathFromHomeDurations is illustratedin FIG. 10.

In one embodiment, a unit 8 number format is used to store a_(j),v′_(i)[j], or w. In one embodiment, the unit of a number stored isminutes. In one embodiment, a sum v′_(i)[j]+w, for some j, is performedusing Saturation Arithmetic in the range from 0 to a threshold, forexample 255. In one embodiment, a summand v′_(i)[j] is first convertedinto a wider number format, for example into an fp16 format or a unit 16format, and only then added to w, to avoid an arithmetic overflow duringan addition. In one embodiment, some calculations included in themathematical formula are performed using at least one instruction forVector Operations (including tensor operations), for example an AVX-512instruction or a GPU intrinsic supported by hardware. In one embodiment,some calculations included in the mathematical formula are partitioned,and the partitions are executed in parallel.

In one embodiment, the method uses a nearest-neighbor data structure,for example a KD-tree, on stopstation cluster locations to quicklycompute the set A during request processing.

In one embodiment, the method limits the set A to at most a certainnumber of stopstation clusters that are nearest W₁, for example at most100.

In one embodiment, the method limits the set A to stopstation clusterswithin a walk of at most a certain length to W₁, for example 2000meters.

In one embodiment, the method precomputes shortest durations of walksbetween points within a threshold distance from each home clusterlocation and the home cluster location, or precomputes shortestdurations of walks between points within a threshold distance fromstopstation cluster locations and the stopstation cluster location.Then, during request processing, the method does not query any walkengine, but instead uses the precomputed walk durations.

In one embodiment, a shortest duration of a walk is estimated using ageodesic line that ignores any obstacles. This may speed up thecomputation of a walk(s_(i)→W₁) at the expense of accuracy.

In one embodiment, travel paths are used in Equation 1. For example, ifa user request specifies a condition that limits the number of vehicletransfers, we filter s_(i) and j in the equation: we look up the numberof transfers stored in a travel path, and when that number exceeds thelimit, we ignore the specific s_(i) and j in the equation.

In one embodiment, grouping travel durations by the stopstation s_(i),just like we did in v′_(i), improves data access performance. Indeed,although the set A depends on a user request that is not known inadvance, for each s_(i) in the set A, predictably a large number oftravel durations need to be accessed.

In one embodiment, the vectors v′_(i), for a collection of i, aredistributed randomly across different processing units. This maydecrease the latency of computing the vector (a₁, . . . , a_(m)),because the stopstations included in the set A will often be evenlydivided across the processing units.

In one embodiment, the vectors v′_(i), for a collection of i, aregrouped geographically inside a processing unit. This may increase thethroughput of computing the vector (a₁, . . . , a_(m)), because of areduced need for data transfer, due to the fact that the set A oftenconsists of stopstations located near one another.

4.5.3 End Part: Travel W_(k)→H

The computation is analogous, but uses vectors v, not v′. See FIG. 9 foran illustration. The method computes a set B of stopstation clusterswithin a threshold distance from W_(k) (in FIG. 9 the B={s₄, s₅}). Thenthe method computes a shortest travel duration to the home cluster h_(j)as

$\begin{matrix}{b_{j} = {\min{\left\{ {{{walk}\left( W_{k}\rightarrow H_{j} \right)},{\min\limits_{s_{i} \in B}\left\{ {{{walk}\left( W_{k}\rightarrow s_{i} \right)} + {\upsilon_{i}\lbrack j\rbrack}} \right\}}} \right\}.}}} & \left( {{Equation}\mspace{14mu} 2} \right)\end{matrix}$

Similar as before, in one embodiment, because of the vectorrepresentation of the v, the method jointly computes a travel durationto every home cluster using vector operations on the v according toEquation 2. In one embodiment, the method uses a mathematical formula

$\min\limits_{s_{i} \in B}{\left\{ {{{walk}\left( W_{k}\rightarrow s_{i} \right)} + \upsilon_{i}} \right\}.}$

The vector (b₁, . . . , b_(m)) of travel durations is denoted byPathToHomeDurations(W _(k)→)=(b ₁ , . . . ,b _(m)).

The method uses embodiments similar as in Section 4.5.2. For example, inone embodiment, the method uses a vector operation that togethercomputes a part of the vector (a₁, . . . , a_(m)) and a part of thevector (b₁, . . . , b_(m)).

4.5.4 Combining Start, Middle, and End Parts

Finally, the method computes the shortest travel duration of the commutepath for every home cluster. The method simply adds the two vectors andshifts values at the coordinates. We denote the resulting vector byPathDurations(→W ₁ →W ₂ → . . . →W _(k)→)=PathFromHomeDurations(→W₁)+PathToHomeDurations(W _(k)→)+PathNonHomeDuration(W ₁ →W ₂ → . . . →W_(k)),  (Equation 3)where the first “+” is a coordinate-wise addition of vectors, and thesecond “+” is an addition of a number to a value at every coordinate ofa vector.

In one embodiment, the method computes travel durations given specificdeparture times from, or arrival deadlines to, any geographical locationalong any commute path.

In one embodiment, the method precomputes PathDurations for specificcommute paths. For example, the method computes and storesPathFromHomeDurations(→W)andPathToHomeDurations(W→),for every W that is a school. During request processing, the methodretrieves precomputed vectors from the database, instead of computingthem with Equation 1, Equation 2, and Equation 3.

In one embodiment, the value of PathDurations may be changed. Consider aschool W that has a zoning requirement that specifies that only certainhomes may send children to the school. This can be simply achieved bysetting the values at coordinates of a vector PathFromHomeDurations (→W)to infinity for these home clusters that are outside of the zone of theschool, and similarly setting values at coordinates of a vectorPathToHomeDurations (W→). A similar change can be made when W is arestaurant with a limited delivery area, or a government office with alimited jurisdiction. Thus changed values may be stored, and retrievedduring request processing.

In one embodiment, our method applies an arbitrary “manipulation”function to the values at coordinates of vectors v_(i), v′_(i), orPathDurations. For example, this can help implement a policy of agovernment that reduces the transportation charge for commuting from orto certain parts of a metropolitan area in a special economic zoneaffected by a disaster. An application can be performed before requestprocessing (for example a global policy), or during request processing(for example a user-personalized policy).

In one embodiment, the method computes PathDurations for a subset ofhome clusters. For example, the subset is determined from a condition onhome features specified by a user request. In one embodiment, we selectthe sequence h₁, . . . , h_(m) of home clusters so that these subsetsoften reside in a short segment of the sequence, which reduces datatransfer.

4.6 Search for One Home

Our method teaches how to efficiently search for a home using commutes.We present a few example search requests first, before we introduce ageneral search request for one home.

4.6.1 Weighted Sum Request

Consider a family with one parent going to work located at geo1 fivetimes per week, and the other parent going to work at geo2 three timesper week. This family wants to find a home with a short weekly travelduration. We can find the weekly travel duration for the family forevery home cluster as a weighted sum of two vectors, as follows:5·PathDurations(→geo1→)+3·PathDurations(→geo2→).

4.6.2 Minimum Request

Consider a single mother who works from home, and is sending a child toschool. The mother wants to find a home that will be near any schoolfrom among schools E. We can find the daily travel duration for thechild for every home cluster as a coordinate-wise minimum of vectors, asfollows:

$\min\limits_{e \in E}{{PathDuration}\mspace{11mu}{\left( \rightarrow\left. e\rightarrow \right. \right).}}$

4.6.3 General Search for One Home

In one embodiment, our method defines a request as any sequence ofcommute paths path₁, . . . , path_(q), for any q, and a functionDeriver:

^(q)→

that maps any vector of q numbers into a number. The method computes thetravel duration vectors (from scratch, by retrieving precomputedvectors, or both)PathDurations(path₁)=(d _(1,1) , . . . ,d _(1,m)),. . . ,PathDurations(path_(q))=(d _(q,1) , . . . ,d _(q,m)),and applies the function Deriver coordinate-wise to the vectors, toproduce a vectorRequestDurations(path₁, . . . ,path_(q),Deriver)=(Deriver(d _(1,1) , . . . ,d _(q,1)),Deriver(d _(1,2) , . . . ,d _(q,2)),. . . ,Deriver(d _(1,m) , . . . ,d _(q,m)))with a “derived” travel duration for each home cluster.

In one embodiment, the function Deriver is a weighted sum of numbersDeriver(x ₁ , . . . ,x _(q))=w ₁ ·x ₁ + . . . +w _(q) ·x _(q),for weights w₁, . . . , w_(q). Weights can be positive or negative. If atravel duration is interpreted as a monetary cost of travel, then anegative weight can be interpreted as a monetary gain, for example ifthe commuter is expected to benefit from performing the specific commute(e.g., a delivery of a parcel). Weights can represent relativeimportance of commute paths, for example a commute path of a ChiefExecutive Officer may have a higher weight than a commute path of afirst-line manager.

In one embodiment, the function Deriver is a minimum of numbers

${{Deriver}\;\left( {x_{1},\ldots\mspace{11mu},x_{q}} \right)} = {\min\limits_{1 \leq i \leq q}{x_{i}.}}$

In one embodiment, the function Deriver is a weighted sum and a minimum

${{Deriver}\left( {x_{1},\ldots\mspace{11mu},x_{r},x_{r + 1},\ldots\mspace{11mu},x_{q}} \right)} = {{w_{1} \cdot x_{1}} + \ldots + {w_{r} \cdot x_{r}} + {\min\limits_{{r + 1} \leq i \leq q}{w_{i} \cdot {x_{i}.}}}}$

In one embodiment, the function Deriver is a conditional, for example

  Deriver(x₁, x₂) = [  if x₁ < 30 then    return x₁ + x₂   else   return ∞],or any algorithm.

4.7 Compare Two or More Homes

Our method teaches how to efficiently compare multiple homes usingcommutes. We illustrate the method on a few examples, before weintroduce a general comparison request.

4.7.1 Current Home

Consider a family that currently lives in a home. The family membershave specific commute paths to work places, schools, and other places.The family considers a move to other home, and wants to compare thetotal travel duration for their current home with the total travelduration for other prospective homes. Our method makes such a comparisonrather simple.

In one embodiment, the method computes the travel duration for everyhome cluster, including the home cluster h_(j) of the current home S ofthe familyRequestDurations=(q ₁ , . . . ,q _(m)).

The method then responds with a “difference” vector: the methodsubtracts the value at the j-th coordinate from the value at eachcoordinate(q ₁ −q _(j) , . . . ,q _(j−1) −q _(j),0,q _(j+1) −q _(j) , . . . ,q_(m) −q _(j)).A negative value at a coordinate of the “difference” vector indicatesthat the home corresponding to the coordinate has a shorter travelduration compared to the current home S.

In one embodiment, the method uses some commute paths for the currenthome, but other commute paths for other homes. Thus the method can helpa family evaluate a what-if scenario: “Suppose we change workplaces, andmove to some other home. How will the new commute duration compare toour current commute duration?”

642 4.7.2 General Comparison for Two or More Homes

Consider a family where the mother and the father must continuecommuting to the same work places (no job change), but the children canchange schools. Consider other example of two families: parents andtheir maternal grandparents. They want to find two homes within around30 minutes from each other, so that one home is near any hospital, andthe other home is near specific schools and work places. Our methodmakes comparisons of such homes with other homes rather simple.

In one embodiment, the method computes travel durations PathDurationsfor home clusters for a range of commute pathsPathDurations(path₁)=(d _(1,1) , . . . ,d _(1,m)),. . . ,PathDurations(path_(q))=(d _(q,1) , . . . ,d _(q,m)).Then the method applies a “generalized” function Deriver that does notoperate coordinate-wise as in Section 4.6.3, but instead operatesjointly on all q·m travel durations, and produces a vector of one ormore numbers. In other words, the generalized function isDeriver:

^(q·m)→

^(y),for some y.

4.8 Search-or-Compare

In one embodiment, the function Deriver is any algorithm, for example arandomized algorithm, that takes any input (for example: the commutepaths, shortest travel paths, shortest travel durations, the homes, orany condition specified by a user request), and produces any output (forexample: a “top list” containing homes that satisfy the user condition,and that have shortest travels that also satisfy the user condition,sorted by the shortest travel durations).

4.9 Variants

Many modifications and variations will be apparent to those of ordinaryskill in the art without departing from the scope and spirit of theembodiments. We present of few variants for illustration.

4.9.1 Extending Travels

In one embodiment, the method first computes incomplete shortesttravels, and then extends them to selected home clusters. This tradesoff additional processing and storage for a decrease in storage andprocessing due to fewer home clusters in vectors v_(i) and v′_(i).

The method determines home connectors which are some elements of thetransportation system. In one embodiment, the home connectors areclusters of homes or clusters of stopstations, which may differ from theclusters discussed in Section 4.4.1. Then the method precomputes andstores in a database shortest travels between the stopstation clustersand the home connectors, using embodiments similar to these of Section4.4.

When a request is received, the method computes shortest travels betweena work W contained in the request, and the selected home clusters. Forthis purpose, the method determines stopstation clusters nearby the workW, and retrieves precomputed shortest travels between the nearbystopstation clusters and the home connectors, to compute shortesttravels between the work W and the home connectors, similar to Section4.5. In one embodiment, this computation uses vector operations similarto these of Sections 4.5.2 and 4.5.3. Then these shortest travels areextended beyond the home connectors, to form shortest travels betweenthe work W and the selected home clusters. This can be achieved bysimply finding, for each home cluster, a minimum extended travel (whichmay be not just a walk) that passes through any home connector near thehome cluster, for example within 2000 meters, and sometimes finding adirect shortest travel between the work W and the home cluster (similarto Sections 4.5.2 and 4.5.3). In one embodiment, in an overlap area themethod uses both the home connectors and the vectors v_(i) and v′_(i).

The performance of extending travels is proportional to the total numberof home connectors, and also proportional to the number of homeconnectors near any selected home cluster. In one embodiment, theextending is used in sparse parts of a metropolitan area, where the twoquantities are likely to be low. In one embodiment, the method uses aperformance cost function to: (1) determine home connectors, (2) selecthome clusters, and (3) determine a subset of nearby home connectors foreach of the selected home clusters.

4.9.2 Commute by Car

In one embodiment, we compute travel durations of commute paths by car,rather than by public transportation. This can be achieved by simplemodifications to earlier sections.

When preprocessing a transportation system for homes, Section 4.4,instead of taking a GT to be a graph of a public transportation system,we take a GT to be a graph of a car drive system. This car drive graphcan be obtained, for example, from prior art. That graph may havevertexes representing geographical locations on roads, and edgesrepresenting driving a car or turning a car; edges may contain dataabout drive durations at various times of the day, for example duringrush hours.

We extend GT. For each home cluster s, we add vertexesHOME_CLUSTER_SOURCE_sandHOME_CLUSTER_TARGET_s.We connect each home cluster with some roads that are within a thresholddistance, for example meters, by adding a vertexCONNECTOR_r,for at least one r. In one embodiment, the vertex denotes ashortest-distance projection of a home cluster location onto a road thatis within a threshold distance; the road may be inside a car parkingassigned to the home cluster. The vertex is connected to the two homecluster vertexes by edges labeled Zero with weight 0. The vertex is alsoconnected to vertexes representing endpoints of a segment of the road.Instead of adding clusters of stopstations, which we should not becausenow GT does not model any subway station nor bus stop, we add vertexesREPRESENTATIVE_SOURCE_s′andREPRESENTATIVE_TARGET_t′for a collection of s′ and t′; these vertexes represent locations thatfrequently occur in shortest travels; the locations can be obtained, forexample, using prior art. These vertexes may represent clusters. Thesevertexes get appropriately connected with other vertexes using edges.

We use the extended GT to compute shortest durations of graph paths fromeachREPRESENTATIVE_SOURCE_s′to everyHOME_CLUSTER_TARGET_s,and from eachHOME_CLUSTER_SOURCE_sto everyREPRESENTATIVE_TARGET_t′,possibly using embodiments similar to these described in Section 4.4.1.These travel durations will be stored in a vector form v_(i) and v′_(i)as described in Section 4.4.2.

When computing a travel duration of a commute path, we use car rides,instead of using walks as in Section 4.5.

For a start part of a travel, we make the following modifications toSection 4.5.2. Instead of walk(H_(j)→W₁), we use a duration of a carride from a home cluster H_(j) to W₁, for example obtained from priorart; in one embodiment we instead set this duration to infinity when thedistance between H_(j) and W₁ is above a threshold. The set A is a setof vertexesREPRESENTATIVE_TARGET_t′that are within a threshold distance from W₁. For each vertex s_(i)∈A,we compute a car drive duration from s_(i) to W₁, for example usingprior art. Then we apply these modifications to Equation 1.

For an end part of a travel, we make similar modifications, but now toSection 4.5.3. Instead of walk(W_(k)→H_(j)), we use a duration of a carride from W_(k) to a home cluster H_(j), for example obtained from priorart; in one embodiment we instead set this duration to infinity when thedistance between W_(k) and H_(j) is above a threshold. The set B is aset of vertexesREPRESENTATIVE_SOURCE_s′that are within a threshold distance from W_(k). For each vertexs_(i)∈B, we compute a car drive duration from W_(k) to s_(i), forexample using prior art. Then we apply these modifications to Equation2.

We use other embodiments stated in Section 4.5. For example, we mayestimate a travel duration of a car ride using a geodesic.

The search framework of Section 4.6 can be simply extended to allow torestrict what kind of vehicles should be used for each commute path. Asa result, for example, we can determine travel durations of commutepaths for every home for a family with one parent going to work locatedat geo1 by car five times per week, and the other parent going to workat geo2 by public transportation three times per week.

The comparison framework of Section 4.7 can similarly be extended. As aresult, for example, we can search for a new home from which a personwill commute by car, and compare with the current travel duration bypublic transportation. A result of our method may be used as anincentive to purchase a car due to a move.

4.9.3 Commute by Other Means

In one embodiment, the method uses commute paths by other means, forexample: walk only; bicycle only; express buses and walks only; subwayand walks only; express buses, subway and walks only; shared vans andwalks only; boats; airplanes; and so on. We simply use modificationssimilar to these described in Section 4.9.2. In one embodiment, themethod identifies other home and other means of travel, and recommendsthese to the user, explaining gains compared with the current home andthe current means of travel.

Shortest graph paths in a given graph can be computed without knowinggeographical locations of the various vertexes of the graph. Hence, inone embodiment, the method uses a transportation system whose variouselements lack geographical location.

The transportation system need not physically move objects. The methodmerely needs to be able to determine route or route lengths between theelements of the transportation system. Thus a computer network, thatmoves data, is an example of a transportation system, comprising thesetransportation elements: wires/lines (analogous to roads), andhubs/switches (analogous to stops/turns). Many other examples of atransportation system will be apparent to anyone of ordinary skill inthe art.

4.9.4 Conditions on Travel Paths

We can simply realize various filtering conditions on travel paths. Forexample, we can build a graph G that has no transfer between any subwaylines but has walks between home clusters and station clusters. Alongwith modifications similar to these described in Section 4.9.2, themethod will then search-or-compare homes when a commuter may sit all theway during a ride between home and work (both the home and the work willneed to be within a walking distance from a subway stop of one subwayline). Similarly, we can build a G and modify shortest graph pathalgorithms so as to restrict graph paths to at most one transfer, or asubway-bus transfer, or a bus-subway transfer, or a transfer that occurswithin a time window, or a type of travel path commonly used bycommuters. Other variations for filtering travel paths fall within thescope of our method, as will be apparent to anyone of ordinary skill inthe art.

4.9.5 Conditions on Homes

In one embodiment, the method receives a filtering condition on variousfeatures of homes, and searches or compares homes whose features matchthe condition. The features may be: a type (e.g., a detached home or ahigh-rise apartment), a transaction type (e.g., sale or rent), a price,a real estate agent commission fee, taxes, a maximum bank loan amount, anumber of bedrooms or bathrooms, an area/size of a home, geographicaldirections of windows, a floor number of a home, the number of floors inthe building, or a typical monthly management fee. Simply the methodmaintains a feature list for each home, and, given a condition,determines the homes whose features match the condition. The traveldurations for these matching homes can be obtained from travel durationsfor home clusters. Other variations for filtering homes fall within thescope of our method, as will be apparent to anyone of ordinary skill inthe art.

4.9.6 Meta Search-or-Compare

In one embodiment, the method searches or compares homes using anensemble of prior search-or-compare requests and responses. This can beviewed as a meta method (a method that uses itself). It is useful, forexample, for estimating the value of a proposed new real estatedevelopment.

We describe an embodiment of the meta method. We receive some number uof commute paths path₁, . . . , path_(u). In one embodiment, the commutepaths come from a usage log of a computer service for searching orcomparing homes, each commute path may be provided by a different userof the service. For each commute path path_(k), we compute the traveldurations for all home clusters PathDurations(path_(k)) according toSection 4.5. Then, we apply an aggregator that processes the traveldurations. In one embodiment, for each commute path path_(k), we receivea vector of weights w_(k). In one embodiment, each weight is aprobability of a user clicking on a home in the corresponding homecluster, which may be affected by a condition for a real estate propertyor for a travel path specified in the user request. Then we computeaggregates coordinate-wise, as in the following formula

$\frac{1}{u}{\sum\limits_{k = 1}^{u}\;{{w_{k} \cdot {PathDurations}}\mspace{11mu}{\left( {path}_{k} \right).}}}$The j-th aggregate is an average weighted travel duration for homecluster H_(j). In one embodiment, the j-th aggregate denotes anormalized contribution of the home cluster to the total travel durationof the entire metropolitan area; in a sense, it is a centrality of thehome cluster within the metropolitan area. Because our method rapidlycomputes PathDurations(path_(k)), we can compute the aggregates quickly.This enables a rapid computation of a desirability of each real estateproperty with respect to commuting.

Many other embodiments of the meta method will be apparent to those ofordinary skill in the art. In other embodiment, the commute paths aregenerated from geographical locations of homes and works. In otherembodiment, the weights are set to non-zero for some number of homeclusters with the lowest travel durations, others are set to zero. Inother embodiment, a data scientist evaluates what-if scenarios; becauseof advantages of our method, these scenarios can be evaluated rapidly.In other embodiment, the aggregator is an arbitrary algorithm, forexample one that computes a variance, quantiles, a cumulativedistribution function, or a probability of exceeding a threshold.

4.9.7 Commute Path Involving Two or More Homes

In one embodiment, a commute path involves any two or more homes. Thisis useful, for example, when searching or comparing homes jointly fortwo or more families.

In one embodiment, H₁ and H₂ are two home clusters. We can compute atravel duration of any commute path H₁→path→H₂ that starts at H₁, butends at H₂ that may be different from H₁. We simply use a jcorresponding to H₁ in Equation 1, but use a j corresponding to H₂ inEquation 2.

In one embodiment, we precompute both vectors (a₁, . . . , a_(m)) and(b₁, . . . , b_(m)), and then we find a travel duration for arbitrarytwo home clusters H_(i) and H_(j) by simply adding a value at acoordinate of (a₁, . . . , a_(m)) corresponding to H_(i) to a value at acoordinate of (b₁, . . . , b_(m)) corresponding to H_(j), plus thetravel duration of the middle part of the path (which is independent ofH_(i) and H_(j)). We remark that a travel duration for any pair of homeclusters, from among the m² pairs, can be found in constant O(1) timeusing only linear O(m) space. This is because of an additive structureof the travel duration and the fact that the path acts as a separatorbetween any two home clusters.

In one embodiment, the set of home clusters allowed for H₁ may bedifferent from the set of home clusters allowed for H₂. For example, auser condition can restrict H₁ to be in the east part of a city, whileH₂ to be in the west part of the city. In this case we may use a vector(b₁, . . . , b_(m′)) of length at m′ that is different from m.

More generally, the method computes a travel duration for any sequenceof k≥2 home clusters H₁→path₁→H₂→path₂→ . . . →path_(k−1)→H_(k). Usingprecomputing, this can be done in O(k) time and O(m·k) space.

In one embodiment, a commute path includes a direct travel between homeclusters, H_(x)→H_(y), as in this example commute path:H₁→path₁→H_(x)→H_(y)→path₂→H₂. In this example, our method partitionsthe computation into three parts: (1) compute a travel duration ofH₁→path₁→H_(x), (2) compute a travel duration of H_(x)→H_(y), and (3)compute a travel duration of H_(y)→path₂→H₂. The parts (1) and (3) canbe performed using the method described earlier in the current Section4.9.7. The part (2) can be performed similarly, conceptually treatingeither H_(x) or H_(y) as a work. For example, using the vectors at thecoordinate corresponding to H_(x), we obtain travel durations from thehome cluster H_(x) to every stopstation cluster; these can be retrievedas a column of the precomputed vectors v′_(i), 1≤i≤n. Then we extendfrom each stopstation cluster within a threshold distance from H_(y) toform a complete travel to H_(y), similar to Section 4.5.2, see forexample FIG. 8 and Equation 1. Because of precomputing due to ourmethod, the (3) can be performed in time proportional to the number ofthe stopstation clusters within the threshold distance. In otherembodiment, because the home clusters are known in advance, weprecompute a travel duration between every pair of home clusters, andthen simply retrieve a travel duration in constant O(1) time. Naturally,a similar partitioning applies to any commute path that contains one ormore direct travel between home clusters.

4.9.8 Space Exploration

Our method introduces high-performance space exploration algorithms.Consider a case when a commuter travels along a specific commute pathH₁→path→H₂ which has a specific cost; the cost function depends on thetwo homes H₁ and H₂. The commuter seeks homes H₁ and H₂ that minimizethe cost. This is useful, for example, when a family is willing tochange work locations (H₁ and H₂) of the two parents at the same time,while minimizing the total travel duration from the current homelocation (inside path) of the family, aka the two-body problem.

One embodiment of space exploration is a gradient descent algorithm. Thecost function can be any differentiable function; for example, given afixed path, the function takes two homes H₁ and H₂ as an input, andreturns as an output a distance between the two homes H₁ and H₂multiplied by a travel duration of a commute path H₁→path→H₂ (thisfunction can be appropriately extended outside of the discrete domain ofhome pairs, for example via geographical location of homes andextrapolation). At a given step of the gradient descent algorithm, thealgorithm computes a gradient of the cost function given a pair ofhomes, and then picks a pair of homes along the direction of thegradient for the next step of the algorithm. The gradient can becomputed using values of the cost function, for example using thetwo-point formula that needs two values. In one embodiment, the outputof the gradient descent algorithm is a pair of homes where the gradienthas a small enough norm.

In one embodiment, we precompute the vectors (a₁, . . . , a_(m)) and(b₁, . . . , b_(m)) consuming linear O(m) space, and at each step ourmethod computes the gradient in constant O(1) time. Thus the gradientdescent algorithm can often make rapid progress.

In one embodiment, we do not precompute (a₁, . . . , a_(m)) and (b₁, . .. , b_(m)), but instead produce the needed values per Equation 1 andEquation 2 on-demand. Such approach may be used when the number of homesis so large that precomputing (a₁, . . . , a_(m)) or (b₁, . . . , b_(m))is infeasible.

In one embodiment, a space exploration algorithm may have constraints onhomes. For example, if we require a round-trip commute to one home, aconstraint will be H₁=H₂. We may restrict a geographical region allowedfor H₁, or allowed for H₂.

In one embodiment, a space exploration algorithm uses a commute pathinvolving only one home: H₁→path or path→H₁. In other embodiment, aspace exploration algorithm uses a commute path involving two or morehomes, as in Section 4.9.7.

In one embodiment, a cost function depends on homes and works. Forexample, such cost function can be used when a person seeks to minimizea blend of expended time and money. In one embodiment, the cost functionis: the travel duration between H and W, plus the monetary cost ofrenting H, minus the salary paid by W, possibly with weights thatrepresent relative importance of the constituents. In one embodiment,because of advantages of our method, a gradient descent can efficientlybe used to search for homes and works under such cost function.

4.10 General Case

In one embodiment, the words “home” and “work” have an arbitrarysemantic meaning. For example, consider a case where a person is lookingfor work that is located near the current home of the person. The persondoes not want to move to a different house, but merely wants to find ajob closer to the current home. In this case, our method can be simplyapplied. Given a range of sites where people work throughout ametropolitan area (for example various offices, factories, etc.), themethod computes travel durations between every work site cluster, andeach stopstation cluster. Thus the method can be viewed as a “worksearch-or-compare using commute” method. In one embodiment, the methodsearches or compares work sites based on a user-specified job type, or asalary range, and based on a travel duration from the current home ofthe user. As other example, consider a corporation that wants to moveits headquarters to a new location. Our method can be used to computethe total “corporate travel duration” for every new location of theheadquarters throughout the metropolitan area. Thus, the corporation maydetermine how each new location will affect commutes of the employees. Anew location may be selected, for example, so that: (1) the worst-casecommute duration is limited, and (2) the average commute duration islow; thus satisfying individual and social objectives.

Our description so far mainly talked about a travel duration as thesearch or compare objective. However, the method can use any otherobjective, for example: a monetary cost of travel; a metric distance;specific features or attributes of travel paths, for example: the numberof transfers, or a walking distance; or features of homes, for example:a price, a size, or a type. In one embodiment, this can be simplyachieved by building graphs and setting edge weights appropriately.Various objectives may be combined into a multi-objective optimizationsearch based on a multi-dimensional cost, for example to search for ahome that minimizes a travel duration that is penalized by the monetarycost of travel.

In general, the method uses arbitrary sites S₁, . . . , S_(m) (a sitewas called a home in earlier sections) and arbitrary places P₁, . . . ,P_(k) (a place was called a work in earlier sections), and the methodsearches or compares the sites S₁, . . . , S_(m) using routes or routelengths (in earlier sections: a route was called a description oftravel, and a route length was called a length of travel) that originateor end at some of the sites and visit some of the places, as specifiedby arbitrary route specifications (a route specification was called acommute path in earlier sections) involving the sites and the places.The searching or comparing may use any variant described in Section 4.9.Thus the method may respond with routes or route lengths, or theirrepresentation. Information computed by one embodiment of the method canbe used, recursively, as an input to any embodiment of the method.

4.11 Computer System

One of the embodiments of the invention is a computer system thatsearches or compares real estate properties using commutes. Weillustrate an embodiment of the computer system in FIG. 11.

We use the term “module” in our description. It is known in the art thatthe term means a computer (sub)system that provides some specificfunctionality. Our choice of partitioning the computer system into thespecific modules is exemplary, not mandatory. Those of ordinary skill inthe art will notice that the system can be organized into modules inother manner without departing from the scope of the invention.

In one embodiment, each travel along any commute path has a specificdeparture time.

One module (1101) of the system reads data about the transportationsystem from a data source (1102), and constructs the graph G. During theconstruction, the module retrieves data about homes from a real estatedata source (1103), and retrieves from other data source (1104) shortestwalks between and near home clusters and stopstation clusters. The graphG contains timing data about vehicles. The module outputs the graphwithout anyHOME_CLUSTER_SOURCE_svertexes (1105), and also outputs the graph without anyHOME_CLUSTER_TARGET_svertexes but with reversed edges (1106). The module also builds anearest neighbor data structure (1107) that can find stopstations withina threshold distance from any given geographical location, andprecomputes shortest walks near home clusters and stopstation clusters(1108).

In the meantime, other module (1109) of the system reads the two graphs,and computes shortest graph paths. The module considers a range of timesduring the day, in one embodiment every 5 minutes. For each departuretime, the module generates one table (1110) with shortest traveldurations from stopstation clusters to home clusters using (1105), andthe other table (1111) with shortest travel durations from home clustersto stopstation clusters using (1106). In one embodiment, each travelduration, rounded to the nearest minute, is stored as the unit 8_t typeof the C++ programming language, with the maximum value of 255 reservedto denote an unknown or too large travel duration. In one embodiment,the tables are laid out on HDD disks in the row-major order. In oneembodiment, the system uses a cache hierarchy involving HDD disks, SSDdisks, and the main memory. In one embodiment, the tables or their partsare compressed using any compression algorithm, for example deltacompression. We observe that for any pair comprising a home cluster anda stopstation cluster, travel durations are often similar during aperiod of time. This similarity often also holds in a neighborhood ofthe pair. In one embodiment, we select the sequence h₁, . . . , h_(m) ofhome clusters so that any h_(i) and h_(i+1) that are adjacent in thesequence often are nearby home clusters, or select the sequence s₁, . .. , s_(n) of stopstation clusters so that any s_(i) and s_(i+1) that areadjacent in the sequence often are nearby stopstation clusters.

The modules (1101) and (1109) operate continuously. As a result, thesystem maintains fresh data about travel durations given departuretimes.

Concurrently, a path durations module (1112) computes PathDurations.Given a commute path with departure times, the module queries (1113) forany relevant PathDurations that has already been precomputed. Anymissing one is computed from scratch: the module queries a navigationdata source (1114) to compute the PathNonHomeDuration of the part of thecommute path that does not involve any home. The module also computesthe travel durations PathFromHomeDurations and PathToHomeDurations thatinvolve homes, by querying the nearest stopstations (1107), walks(1108), and home travel duration vectors (1110 and 1111) at thedeparture times.

Concurrently, the request processing module (1115) searches or comparesreal estate properties. Any request (1116) contains commute pathsincluding geographical locations along the commute paths and departuretimes, and a Deriver. When a request is received from a user, the moduleretrieves PathDurations from the path durations module (1112), appliesthe Deriver, and responds to the user with information representing anoutput of the Deriver (1117).

Aspects of the invention may take form of a hardware embodiment, asoftware embodiment, or a combination of the two. Steps of theinvention, for example blocks of any flowchart, may be executed out oforder, partially concurrently or served from a cache, depending onfunctionality or optimization. Aspects may take form of a sequentialsystem, or parallel/distributed system, where each component embodiessome aspect, possibly redundantly with other components, and componentsmay communicate, for example using a network of any kind. The inventionis not described with reference to any specific programming language. Acomputer program carrying out operations for aspects of the inventionmay be written in any programming language, for example C++, Java, orJavaScript. Any program may execute on an arbitrary hardware platform,for example a Central Processing Unit (CPU) or a Graphics ProcessingUnit (GPU), and associated memory or storage devices. A program mayexecute aspects of the invention on one or more software platforms,including, but not limited to: a smartphone running Android or iOSoperating systems, or a web browser, for example Firefox, Chrome,Internet Explorer, or Safari.

4.12 Computer Service

One of the embodiments of the invention is a computer service forsearching or comparing real estate properties using commutes. Theservice is available to users through a user-accessed device, forexample a smartphone application or a webpage. It will be obvious toanyone of ordinary skill in the art that the invention is not limited tothese devices. It will also be obvious that the presentation of theservice in our drawings can be modified (for example by rearranging,resizing, changing colors, shape, adding or removing components) withoutdeparting from the scope of the invention.

In one embodiment, the service is accessed through a smartphoneapplication. See FIG. 12 for an illustration. The user inputs a request.In one embodiment, the request includes:

-   the desired features of a real estate property (1201), for example    “3 bedroom, tall building, high floor”;-   the commute paths (1202, 1203, “schools Towsend or Jericho High”    1204), departure times (1205, 1206), and the frequency of each    commute path (1207, 1208); and-   the geographical location of the current home (1209) of the user.

In response, the service returns information representing traveldurations. For example, the service renders geographical locations ofreal estate properties that match the user request (1210). The servicerenders how the travel durations from these real estate propertiescompare to the travel duration from the current home. The servicerenders a summary about each matching real estate property, for exampleits price. The real estate properties may be stacked on a 2D map, sothat a real estate property with a lower travel duration appears abovethese real estate properties with higher travel durations. When there isclutter on a map, the service may instead render a cluster of realestate properties, the size of which is correlated with the number ofreal estate properties in the cluster. The cluster may display asummary, for example the number, or typical features of the real estateproperties in the cluster.

In one embodiment, the service renders real estate properties with theshortest travel durations. Summaries of the real estate properties arealso rendered (1211). The real estate properties may be sorted by thetravel duration.

In one embodiment, the service renders a “heatmap” that uses a color toindicate a travel duration from each region of a metropolitan area, forexample a minimum for the region, see FIG. 1 for an illustration. Aheatmap may depict a difference between a travel duration for any homeand the travel duration for the current home of the user. In oneembodiment, a heatmap renders only the real estate properties that matchthe user-desired features of a real estate property.

In one embodiment, the service renders a histogram of travel durations(1212). The histogram has the travel duration on one axis, and on theother axis the fraction of real estate properties that yield this travelduration. In one embodiment, a histogram renders only the real estateproperties that match the user-desired features of a real estateproperty. In one embodiment, the user can scroll (1213) through ahistogram to any section of the histogram, and the service rendersresults for travel durations of this specific section. In oneembodiment, the method uses other form of a histogram, for example a piechart.

In one embodiment, the user may restrict the travel paths, for exampleby limiting a total walk duration, a number of transfers, etc., forexample using range sliders (1214).

In one embodiment, the service renders a summary of a travel path for areal estate property.

In one embodiment, the service responds to the user with at least oneof:

-   (a) a geographical location of: a place, or a site, rendered on a    map;-   (b) a departure time or an arrival time for: a place, or a site;-   (c) a summary of a site; a summary may include at least one of: a    name or an address of the site, a price of the site, or a size of    the site;-   (d) a summary of sites that form a cluster of nearby sites;-   (e) a rendering of the cluster of nearby sites, a size of the    rendering being correlated with a number of sites in the cluster of    nearby sites;-   (f) a stacking of sites in a z-index order of a route length,    shorter route length higher up the stacking, rendered on a map;-   (g) information about a route or route length; information may    include at least one of: (i) a route: length, duration, monetary    cost, speed, or wait duration; (ii) a name of: a vehicle, a vehicle    road, a walking path, a stop, a turn, or a transit station of a    public transportation vehicle; or (iii) a geographical location of:    a monetary cost, a speed, a wait duration, a vehicle, a vehicle    road, a walking path, a stop, a turn, or a transit station of a    public transportation vehicle;-   (h) a histogram of a route length across the sites; or a heatmap of    a route length across the sites rendered on a map;-   (i) a minimum route length; a histogram of a minimum route length    across the sites; or a heatmap of a minimum route length across the    sites rendered on a map;-   (j) a weighted route length; a histogram of a weighted route length    across the sites; or a heatmap of a weighted route length across the    sites rendered on a map;-   (k) a difference route length; a histogram of a difference route    length across the sites; or a heatmap of a difference route length    across the sites rendered on a map;-   (l) a rendering of an output returned by the Deriver;-   (m) a site that minimizes a route length across the sites, under a    limit on a monetary cost of travel;-   (n) one of the above items (a) to (m) constrained by a condition    specified by a user; or-   (o) a top list of sites, from among these sites, routes, and route    lengths that satisfy the condition, sorted by a route length.

4.13 Claims

Those of ordinary skill in the art shall notice that variousmodifications may be made, and substitutions may be made withessentially equivalents, without departing from the scope of the presentinvention. Besides, a specific situation may be adapted to the teachingsof the invention without departing from its scope. Therefore, despitethe fact that the invention has been described with reference to thedisclosed embodiments, the invention shall not be restricted to theseembodiments. Rather, the invention will include all embodiments thatfall within the scope of the appended claims.

4.14 Glossary

We include a glossary of selected phrases that occur in the claims, andexample references to the specification. These references are notintended to be exhaustive; other references exist. The selected phrasesfollow the order in which the phrases first appear in the claims.

The claims also use the following phrases, whose meaning we explain:

-   1. a phrase “at least one A” is equivalent to “#_(A)≥1, wherein    #_(A) is a number of A”;-   2. a phrase “one or more A” is equivalent to “#_(A)≥1, wherein #_(A)    is a number of A”;-   3. a phrase “plurality of As” is equivalent to “#_(A)≥2, wherein    #_(A) is a number of A”;-   4. a phrase “at least two As” is equivalent to “#_(A)≥2, wherein    #_(A) is a number of A”;-   5. a phrase “one of: A, or B” is equivalent to “#_(A)+#B=1, wherein    #_(A) is a number of A, and #_(B) is a number of B”;-   6. a phrase “at least one of: A, or B” is equivalent to    “#_(A)+#_(B)≥1, wherein #_(A) is a number of A, and #_(B) is a    number of B”; and-   7. a phrase “at least one B or C” is equivalent to “at least one A,    wherein each A is (B or C)”.

phrase example reference method paragraph [003] searching or comparingparagraph [002] site paragraph [148] route or route length paragraphs[013], [127] place paragraph [148] transportation system paragraph [022]representative paragraphs [022], [117] storing paragraph [022] databaseparagraph [022] precomputed Section 4.4 description of travel paragraph[020] receiving paragraph [003] request paragraph [022] start placeSection 4.5.2 end place Section 4.5.3 computing paragraph [089] nearbyrepresentative paragraphs [068], [120], [121] length of travel paragraph[020] threshold paragraph [035] retrieving paragraph [022] not nearbysite paragraph [068] responding paragraph [003] information paragraph[003] representation paragraph [148] real estate property/commutedestination paragraph [002] departure time/arrival deadline paragraphs[066], [090] place P_(i), start place P₁, end place P_(k) Definition 1first site, last site Section 4.9.7 involves Section 4.9.7 site-placeroute or route length paragraph [091] cluster site/clusterrepresentative paragraphs [035], [036] site connector paragraph [112]compression paragraph [153] manipulation function paragraph [093] zoneparagraph [092] transportation elements paragraph [127] segment lengthparagraph [033] graph/graph vertexes paragraph [033] transportationelement vertexes Section 4.9.3 site vertex/representative vertexparagraph [117] graph edge paragraph [033] graph edge sourcevertex/graph edge paragraph [033] target vertex/graph edge weight graphpath or graph path length paragraph [033] time paragraph [042] algorithmparagraph [033] reversed paragraph [040] coordinate paragraph [056]vector v, value v[i] paragraph [056] vector v′, value v′[i] paragraph[058] laying out paragraph [153] list/hash map paragraphs [062], [063]weight w_(i) paragraph [059] lower bound lb_(i), upper bound ub_(i),paragraph [060] scaling factor sf_(i) +operation, min operationparagraph [072] first mathematical formula/second paragraphs [072],[086] mathematical formula partitions paragraph [075] routespecification L_(i) paragraphs [028], [148] deriver paragraph [109]minimum route length Section 4.6.2 weights w₁, . . . , w_(q) paragraph[099] weighted route length Section 4.6.1 site S paragraph [105]difference route length Section 4.7.1 condition Sections 4.9.4 , 4.9.5filtering paragraph [006], Sections 4.9.4, 4.9.5 aggregator paragraph[131] centrality paragraph [131] cost function paragraph [140]exploration algorithm Section 4.9.8 differentiable function paragraph[140] gradient descent paragraph [140] multi-objective optimization/paragraphs [050], [147] multi-dimensional cost computer system Section4.11 apparatus Section 4.12 summary/rendering/stacking paragraph [160]heatmap paragraph [162] histogram paragraph [163] top list paragraph[109]

The invention claimed is:
 1. A method for searching or comparing atleast one site using at least one description of travel between the atleast one site and at least one place within a transportation system,the method comprising: (a) receiving at least one representativeincluded in the transportation system, wherein each representativecorresponds to a vertex of a graph that models the transportationsystem, and wherein a number of representatives is at most a number ofvertexes of the graph multiplied by a predetermined ratio; (b) receivinga request comprising the at least one place; and (c) responding to therequest with information obtained using the at least one description oftravel; the method characterized by: (d) determining at least oneprecomputed description of travel and storing it in a database, whereineach first precomputed description of travel describes travel within thetransportation system between a first site included in the at least onesite and a first representative included in the at least onerepresentative; and (e) computing the at least one description of travelthat comprises a description of travel within the transportation systembetween an endpoint site included in the at least one site and anendpoint place included in the at least one place, wherein the endpointsite or the endpoint place is not included in the at least onerepresentative, and wherein the computing comprises: i. determining atleast one nearby representative included in the at least onerepresentative, wherein a length of travel between each nearbyrepresentative and the endpoint place is within a first threshold; ii.retrieving from the at least one precomputed description of travel asecond precomputed description of travel between the endpoint site andthe nearby representative; and iii. determining a description of travelwithin the transportation system between the nearby representative andthe endpoint place.
 2. A computer system for searching or comparing atleast one site using at least one description of travel between the atleast one site and at least one place within a transportation system,the computer system comprising: (a) one or more processors; (b) anon-transitory computer-readable storage medium storing one or moreprograms for execution by the one or more processors; and (c) the one ormore programs comprising instructions to be executed by the one or moreprocessors so as to perform at least one step; wherein the at least onestep comprises: (d) receiving at least one representative included inthe transportation system, wherein each representative corresponds to avertex of a graph that models the transportation system, and wherein anumber of representatives is at most a number of vertexes of the graphmultiplied by a predetermined ratio; (e) receiving a request comprisingthe at least one place; and (f) responding to the request withinformation obtained using the at least one description of travel; thecomputer system characterized by the at least one step furthercomprising: (g) determining at least one precomputed description oftravel and storing it in a database, wherein each first precomputeddescription of travel describes travel within the transportation systembetween a first site included in the at least one site and a firstrepresentative included in the at least one representative; and (h)computing the at least one description of travel that comprises adescription of travel within the transportation system between anendpoint site included in the at least one site and an endpoint placeincluded in the at least one place, wherein the endpoint site or theendpoint place is not included in the at least one representative, andwherein the computing comprises: i. determining at least one nearbyrepresentative included in the at least one representative, wherein alength of travel between each nearby representative and the endpointplace is within a first threshold; ii. retrieving from the at least oneprecomputed description of travel a second precomputed description oftravel between the endpoint site and the nearby representative; and iii.determining a description of travel within the transportation systembetween the nearby representative and the endpoint place.
 3. Anapparatus for searching or comparing at least one site using at leastone description of travel between the at least one site and at least oneplace within a transportation system, the apparatus comprising sets ofinstructions configured to perform at least one step comprising: (a)receiving at least one representative included in the transportationsystem, wherein each representative corresponds to a vertex of a graphthat models the transportation system, and wherein a number ofrepresentatives is at most a number of vertexes of the graph multipliedby a predetermined ratio; (b) receiving a request comprising the atleast one place; and (c) responding to the request with informationobtained using the at least one description of travel; the apparatusfurther comprising a user-accessed device comprising: (d) a receiverconfigured to receive from a user the request; and (e) a transmitterconfigured to respond to the user with the information; the apparatuscharacterized by the at least one step further comprising: (f)determining at least one precomputed description of travel and storingit in a database, wherein each first precomputed description of traveldescribes travel within the transportation system between a first siteincluded in the at least one site and a first representative included inthe at least one representative; and (g) computing the at least onedescription of travel that comprises a description of travel within thetransportation system between an endpoint site included in the at leastone site and an endpoint place included in the at least one place,wherein the endpoint site or the endpoint place is not included in theat least one representative, and wherein the computing comprises: i.determining at least one nearby representative included in the at leastone representative, wherein a length of travel between each nearbyrepresentative and the endpoint place is within a first threshold; ii.retrieving from the at least one precomputed description of travel asecond precomputed description of travel between the endpoint site andthe nearby representative; and iii. determining a description of travelwithin the transportation system between the nearby representative andthe endpoint place.
 4. A method for searching or comparing at least onesite using at least one route between the at least one site and at leastone place within a transportation system, the method comprising: (a)receiving at least one representative included in the transportationsystem, wherein each representative corresponds to a vertex of a graphthat models the transportation system, and wherein a number ofrepresentatives is at most a number of vertexes of the graph multipliedby a predetermined ratio; (b) receiving a request comprising the atleast one place from a user; and (c) responding to the request to theuser with information obtained using the at least one route; the methodcharacterized by: (d) determining at least one precomputed completeroute and storing it in a database, wherein each first precomputedcomplete route describes travel within the transportation system betweena first site included in the at least one site and a firstrepresentative included in the at least one representative; and (e)computing the at least one route that comprises a route within thetransportation system between an endpoint site included in the at leastone site and an endpoint place included in the at least one place,wherein the endpoint site or the endpoint place is not included in theat least one representative, and wherein the computing comprises: i.determining at least one nearby representative included in the at leastone representative, wherein a length of travel between each nearbyrepresentative and the endpoint place is within a first threshold; ii.retrieving from the at least one precomputed complete route a secondprecomputed complete route between the endpoint site and the nearbyrepresentative; and iii. determining a route within the transportationsystem between the nearby representative and the endpoint place.